Group actions in symplectic and contact topology

辛和接触拓扑中的群作用

• 批准号: 261958-2013
• 负责人: Karshon, Yael
• 金额: $ 2.11万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635353
• 关键词: Group; actions; symplectic; contact; topology

Symplectic geometry has its roots in classical mechanics. It is the mathematical structure that underlies the equations of motion of celestial bodies, spinning tops, and mechanical linkages. The field has gone through spectacular progress in recent decades, and deep connections have emerged with other fields of mathematics as well as theoretical physics.Karshon's research focuses on aspects of symplectic geometry that involve symmetries. A "baby example" is the two dimensional sphere with its rotational symmetry. A higher dimensional example is the complex projective plane; symplectically, this can be viewed as a four dimensional ball with a two dimensional sphere sewed along its edge. As envisioned by Felix Klein in his 1872 "Erlanger programm", there is great benefit in studying a geometry through its group of symmetries. The full symmetry group of a symplectic space is always infinite dimensional; however, for many important spaces (such as the two-sphere or the complex projective plane), inside this infinite dimensional group one can find compact finite dimensional subgroups (which can be thought of rotations in multiple dimensions). Karshon's research programme involves the study of symplectic spaces through these finite dimensional symmetries.Current projects involve new classification schemes, recovering symmetries from their underlying "quotient diffeology", and relations with geometric quantization.

辛几何起源于经典力学。它是构成天体、旋转陀螺和机械连杆运动方程基础的数学结构。近几十年来，辛几何领域取得了惊人的进步，并与其他数学领域以及理论物理学产生了深刻的联系。Karshon的研究重点是涉及对称性的辛几何方面。一个“小例子”是具有旋转对称性的二维球体。一个更高维的例子是复射影平面;从符号上讲，这可以看作是一个四维球，沿着它的边缘沿着缝上一个二维球体。正如菲利克斯·克莱因在他1872年的“埃尔朗格纲领”中所设想的那样，通过几何的对称群来研究几何有很大的好处。辛空间的全对称群总是无限维的;然而，对于许多重要的空间（如双球面或复射影平面），在这个无限维群中可以找到紧致的有限维子群（可以认为是多维旋转）。Karshon的研究计划包括通过这些有限维对称性来研究辛空间。目前的项目包括新的分类方案，从其潜在的“商对称性”中恢复对称性，以及与几何量化的关系。